Normalized defining polynomial
\( x^{16} - 4 x^{15} - 1038 x^{14} + 3917 x^{13} + 245444 x^{12} - 216786 x^{11} + 3587876 x^{10} - 145368967 x^{9} - 1562627365 x^{8} + 3884178456 x^{7} - 5170631142 x^{6} - 60744177080 x^{5} + 104223811504 x^{4} + 101334149360 x^{3} + 615788347520 x^{2} - 2334652019200 x + 1480098257920 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(420102507961629136806462464459178014450156250000=2^{4}\cdot 3^{8}\cdot 5^{10}\cdot 7^{8}\cdot 37^{4}\cdot 41^{14}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $947.24$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 5, 7, 37, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{82} a^{8} - \frac{1}{41} a^{7} - \frac{29}{82} a^{6} + \frac{35}{82} a^{5} + \frac{7}{82} a^{4} + \frac{15}{82} a^{3} - \frac{7}{82} a^{2} - \frac{10}{41} a + \frac{8}{41}$, $\frac{1}{164} a^{9} - \frac{33}{164} a^{7} - \frac{23}{164} a^{6} - \frac{5}{164} a^{5} - \frac{53}{164} a^{4} + \frac{23}{164} a^{3} + \frac{12}{41} a^{2} + \frac{29}{82} a + \frac{8}{41}$, $\frac{1}{12136} a^{10} - \frac{9}{6068} a^{9} - \frac{29}{12136} a^{8} + \frac{2039}{12136} a^{7} + \frac{101}{328} a^{6} + \frac{2637}{12136} a^{5} + \frac{3465}{12136} a^{4} + \frac{1323}{6068} a^{3} - \frac{909}{6068} a^{2} + \frac{1203}{3034} a + \frac{756}{1517}$, $\frac{1}{12136} a^{11} + \frac{17}{12136} a^{9} + \frac{1}{328} a^{8} + \frac{849}{12136} a^{7} + \frac{1157}{12136} a^{6} - \frac{2719}{12136} a^{5} + \frac{2353}{6068} a^{4} - \frac{536}{1517} a^{3} - \frac{1465}{3034} a^{2} - \frac{477}{3034} a - \frac{8}{1517}$, $\frac{1}{898064} a^{12} - \frac{1}{898064} a^{11} - \frac{13}{898064} a^{10} + \frac{909}{449032} a^{9} - \frac{121}{449032} a^{8} + \frac{13563}{56129} a^{7} + \frac{23229}{224516} a^{6} + \frac{425077}{898064} a^{5} - \frac{29795}{449032} a^{4} - \frac{10099}{449032} a^{3} + \frac{22611}{112258} a^{2} + \frac{20813}{56129} a + \frac{25664}{56129}$, $\frac{1}{3592256} a^{13} + \frac{1}{3592256} a^{12} - \frac{89}{3592256} a^{11} + \frac{1}{224516} a^{10} + \frac{41}{21904} a^{9} + \frac{181}{48544} a^{8} - \frac{425881}{1796128} a^{7} - \frac{871089}{3592256} a^{6} - \frac{348085}{1796128} a^{5} - \frac{782605}{1796128} a^{4} + \frac{41371}{449032} a^{3} + \frac{4936}{56129} a^{2} + \frac{38245}{224516} a - \frac{7074}{56129}$, $\frac{1}{665057146144320} a^{14} + \frac{1661273}{73895238460480} a^{13} + \frac{185491763}{665057146144320} a^{12} - \frac{3880557563}{166264286536080} a^{11} - \frac{350827031}{41566071634020} a^{10} - \frac{15723778303}{7389523846048} a^{9} + \frac{1513597214591}{332528573072160} a^{8} + \frac{7498574181481}{44337143076288} a^{7} - \frac{152895877873081}{332528573072160} a^{6} - \frac{19688963976283}{332528573072160} a^{5} + \frac{26469431357}{9236904807560} a^{4} + \frac{2577728075545}{5542142884536} a^{3} - \frac{1287922730921}{4156607163402} a^{2} - \frac{103417471724}{692767860567} a - \frac{248550353029}{2078303581701}$, $\frac{1}{47886375717225457208860661670080651336517341986783616970190080} a^{15} - \frac{110120435658665216859416144366678651491294819}{460445920357937088546737131443083185928051365257534778559520} a^{14} - \frac{1864351399603041296857754050503182988354299987670424589}{23943187858612728604430330835040325668258670993391808485095040} a^{13} + \frac{2366552481427993488161307806201778635874553606857933043}{15962125239075152402953553890026883778839113995594538990063360} a^{12} + \frac{1374077699304736497023682862076310846527605737244668979}{184178368143174835418694852577233274371220546103013911423808} a^{11} + \frac{499344375794560871215151886231887838279083969987231878199}{23943187858612728604430330835040325668258670993391808485095040} a^{10} - \frac{10286388551712566234193904757375885242245127089193080655217}{11971593929306364302215165417520162834129335496695904242547520} a^{9} - \frac{188980998903888084197553257583004455725834665601107676173391}{47886375717225457208860661670080651336517341986783616970190080} a^{8} + \frac{10713413881435828726444415483427313771818400009139209506429343}{47886375717225457208860661670080651336517341986783616970190080} a^{7} - \frac{108062299874230519323335510630456176137685986730410519407449}{1197159392930636430221516541752016283412933549669590424254752} a^{6} + \frac{8518859746797971581103649962432405220050657618771431248938081}{23943187858612728604430330835040325668258670993391808485095040} a^{5} + \frac{32156305091701757251711431161729086484825027395973909601497}{1995265654884394050369194236253360472354889249449317373757920} a^{4} - \frac{222919378272180372690756642990791741606882480439964695429523}{598579696465318215110758270876008141706466774834795212127376} a^{3} + \frac{264824333871966576700482411800484173582638946959505551828115}{598579696465318215110758270876008141706466774834795212127376} a^{2} + \frac{14202226972698650668056285131421010215703537746528773094635}{37411231029082388444422391929750508856654173427174700757961} a + \frac{3139477543631195825034291597357121610124045761101386811391}{37411231029082388444422391929750508856654173427174700757961}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{8}$, which has order $512$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 105524452758000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 38 conjugacy class representatives for t16n813 |
| Character table for t16n813 is not computed |
Intermediate fields
| \(\Q(\sqrt{4305}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{105}) \), 4.4.16885645.1, 4.4.3101445.1, \(\Q(\sqrt{41}, \sqrt{105})\), 8.8.577378139308700625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | R | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $37$ | 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.4.2.2 | $x^{4} - 37 x^{2} + 6845$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 41 | Data not computed | ||||||